Prison officials initially denied Phillips his request, citing security and logistical issues, but Kasich reversed the decision Wednesday.
"Ronald Phillips committed a heinous crime for which he will face the death penalty," Kasich said. "I realize this is a bit of uncharted territory for Ohio, but if another life can be saved by his willingness to donate his organs and tissues then we should allow for that to happen."
Phillips wants to donate to his mother and sister, one of whom has a heart condition and the other kidney disease.
Richard Dieter, the executive director of the Death Penalty Information Center, said he had not heard of another death row inmate having been allowed to donate his organs, but others have asked and been denied.
"I think the Governor did the right thing to allow more time for the medical experts, the ethics people to weigh in and discuss this important issue," Dieter said.
Dr. Robert Higgins, the director of the Ohio State University Wexner Medical Center's Comprehensive Transplant Center, said under the best of circumstances, determining whether a donor and a recipient are compatible can take weeks, but Phillips is particularly high risk.
"Who is going to administer the procurement of the organs and how that would be done, in what circumstances that will be conducted, is really a challenging ethical question," he said.
He is now scheduled for execution July 2, 2014.